Your cat has been under stress. He has been in a strange place full of other animals. Now he is in a new home, with a new family and maybe even other pets. While you are ready to make your cat a part of your family, he may need transition time. He may go under a bed or some other hiding space. A few hours or even a couple of days may pass before your feline friend decides to come out from hiding. When he emerges, try not to make any sudden movements. Talk softly to him and allow him to sniff you when he is ready. Place treats nearby to lure him out. When your cat realizes you will not hurt him, he will become comfortable and content in your presence.
Other Pets: To introduce two cats, keep them separated a couple of days, allowing them to smell each other's bedding. Then allow them to sniff each other under a door. The first encounter may be hostile, but allow the cats to work it out. Keep a spray bottle with water ready in case a fight should occur. If the first attempt is unsuccessful, wait a few days and try again.
When introducting a dog and a cat, When the dog is in a calm state and still on a leash, allow the dog and cat to view each other at a distance. You should exude calm but also express firmness and not allow the dog to chase the cat or the cat to scratch the dog. Another human family member should hold the dog leash while you calmly pet the cat, thereby letting the dog know that the cat is part of the family. Most dogs want to please their human companion. If the dog remains fairly calm, allow the dog, while still on a leash, to get close to the cat.
A dog's basic instinct is to chase a cat. Whether the dog chases or not depends on the introduction and on the cat. A cat who is not afraid of dogs and does not run is less likely to provoke a chase. The new dog and your cat should not be alone together for a few weeks the dog may still chase a running cat. Also, if your dog sees the cat outside, he may feel like the cat is fair game for a chase.
It is important to give equal attention to your original pet. Do not ignore him as this may cause resentment of the new pet, and most important is that the original pet too needs to feel loved, even more so because of the new addition to your family.